1. **Problem Statement:** We want to evaluate the integral $$\int x^3 \sqrt{16 - 9x^2} \, dx$$ using trigonometric substitution. 2. **Identify substitution:** The expression under the square root is of the form $$\sqrt{a^2 - u^2}$$ where $$a^2 = 16$$ so $$a = 4$$ and $$u^2 = 9x^2$$ so $$u = 3x$$. 3. **Set substitution:** Use $$u = a \sin \theta$$, so $$3x = 4 \sin \theta$$ which gives $$x = \frac{4}{3} \sin \theta$$. 4. **Find differential:** Differentiating, $$dx = \frac{4}{3} \cos \theta \, d\theta$$. 5. **Rewrite the square root:** Using the substitution, $$\sqrt{16 - 9x^2} = \sqrt{16 - (3x)^2} = \sqrt{16 - 16 \sin^2 \theta} = 4 \cos \theta$$. 6. **Rewrite the integral:** Substitute all parts: $$\int x^3 \sqrt{16 - 9x^2} \, dx = \int \left(\frac{4}{3} \sin \theta\right)^3 \cdot 4 \cos \theta \cdot \frac{4}{3} \cos \theta \, d\theta$$ 7. **Simplify constants:** $$= \int \frac{4^5}{3^4} \sin^3 \theta \cos^2 \theta \, d\theta = \frac{4^5}{81} \int \sin^3 \theta \cos^2 \theta \, d\theta$$ 8. **Use identity:** Express $$\sin^3 \theta$$ as $$\sin^2 \theta \sin \theta = (1 - \cos^2 \theta) \sin \theta$$: $$\frac{4^5}{81} \int (1 - \cos^2 \theta) \cos^2 \theta \sin \theta \, d\theta = \frac{4^5}{81} \int (\cos^2 \theta - \cos^4 \theta) \sin \theta \, d\theta$$ 9. **Substitute:** Let $$u = \cos \theta$$, then $$du = -\sin \theta \, d\theta$$ so $$\sin \theta \, d\theta = -du$$. 10. **Integral in u:** $$\frac{4^5}{81} \int (u^2 - u^4)(-du) = \frac{4^5}{81} \int (u^4 - u^2) du$$ 11. **Integrate:** $$= \frac{4^5}{81} \left( \frac{u^3}{3} - \frac{u^5}{5} \right) + C = \frac{4^5}{81} \left( \frac{\cos^3 \theta}{3} - \frac{\cos^5 \theta}{5} \right) + C$$ 12. **Back-substitute:** Using $$\cos \theta = \frac{\sqrt{16 - 9x^2}}{4}$$: $$= \frac{4^5}{81} \left( \frac{(\frac{\sqrt{16 - 9x^2}}{4})^3}{3} - \frac{(\frac{\sqrt{16 - 9x^2}}{4})^5}{5} \right) + C$$ 13. **Simplify:** $$= \frac{4^5}{81} \left( \frac{(16 - 9x^2)^{3/2}}{4^3 \cdot 3} - \frac{(16 - 9x^2)^{5/2}}{4^5 \cdot 5} \right) + C$$ 14. **Final expression:** $$= (16 - 9x^2)^{3/2} \frac{27x^2 + 32}{1215} + C$$ This completes the integration using trigonometric substitution.